Method and apparatus for producing an electrical property image using a charge correlation matrix

ABSTRACT

An electrical property imaging system includes an array of sensors placed around an object to measure the surface charges thereon as a sinusoidal voltage is applied thereacross. The resulting distribution of charges inside the object are calculated using a stored charge correlation matrix. Electrical property images are produced from the internal charge distribution image.

RELATED APPLICATIONS

[0001] This application is based upon Provisional Application Ser. No.60/221,465 filed on Jul. 26, 2000.

BACKGROUND OF THE INVENTION

[0002] This invention relates to electrical imaging technology, and morespecifically to an apparatus and method for producing high resolutionimages, with accurate values of the electrical properties of the imagedobject.

[0003] Since Roentgen discovered the ability of x-rays to produce ashadowgram of the interior of a sample, there has been a substantialinterest in scientific and engineering circles in technologies thatallow the imaging of the interior of a sample using quantities measuredexterior to the sample. While these technologies are often applied inindustry and commerce generally, one of the most active areas for theuse of imaging technologies is in the field of medicine. The originalshadowgram x-rays were enhanced substantially with the discovery andperfection of computerized axial tomography, which allows the recoveryof not just a shadowgram, but detailed information about the interiorstructure of a sample from x-ray intensities measured on the outside. Asimilar intense interest developed when nuclear magnetic resonancemeasurements were extended to map the interior of a sample in what isnow commonly called Magnetic Resonance Imaging (MRI). Also, the use ofultrasound to explore the interior of samples has been a technology thathas received substantial interest.

[0004] For all of these technologies (x-ray tomography, MRI, andultrasound), an accurate picture in terms of spatial resolution isproduced. However, information about the character of various objectslocated in the interior of a sample is often very limited. For example,x-ray tomography measures only the intensity of absorption at a singlex-ray frequency and is generally simply proportional to the density ofthe material of the sample. The only improvements to conventional x-raytomography have been to use x-rays at different frequencies, which allowsome information about both the density of objects in a sample (in termsof mass) as well as the electron density. Nonetheless, at best, only twonew pieces of information are available from such measurements. MRI is amodality that is sensitive to other parameters. Primarily, MRIs measurethe number of protons (usually associated with hydrogen atoms) at anygiven point in a sample. Some extra information can be obtained with agreat deal of analysis and care by measuring the decay time for certainmagnetic resonance properties, but the sensitivity is such that perhapsthree parameters can be measured using this technique. A similarsituation is obtained with ultrasound where there are yet some othercomplications due to the multiple reflections of the sound waves. WhileCAT scans and MRIs produce pictures that are somewhat familiar to eventhe untrained eye, ultrasound imaging requires a very skilled operatorto perform the measurements and to interpret the results.

[0005] Because of the limitations of the existing imaging techniques,scientists and engineers have looked for other properties that might beexploited to produce an appropriate and improved image of the interiorof an object. Techniques have been developed which measure theelectrical properties of different materials located within an object.Such imaging has shown that substantial variations within a sample fromone type of material to another may be detected (e.g., in a biologicalsample such as a human being, from one type of tissue to another) andprovides a unique imaging modality that reveals information quitedifferent from conventional imaging modalities.

[0006] Such electrical property imaging techniques are often referred toas “impedance tomography.” Most conventional electrical property imagingtechniques are based on the premises that: 1) electrodes, or sensors,should be attached directly to the sample to be measured (for medicalapplications, the sample is a human body), and 2) current is injectedsequentially through each electrode into the sample and the subsequentvoltages measured. Therefore, these conventional imaging techniquesimplement a “constant current/measured voltage” scheme.

[0007] In a departure from such conventional electrical property imagingtechniques, one of the present inventors arranged sensors in an arrayoutside the object to be measured as disclosed in U.S. Pat. No.4,493,039. Further, during imaging of a sample, the ac voltages wereapplied at a fixed amplitude while the current was measured. Thisapproach was further improved as described in pending patent applicationWO 99/12470 by filling the space between the object and the sensor arraywith an impedance matching medium. In addition, techniques for computingthe internal charge distribution based on the measured surface chargesare described, referred to as the scale factor technique and theiterative technique. Both the iterative and scale factor techniquerequire initial estimates of the geometry of internal structures derivedfrom an associated imaging system such as an x-ray CT system. Theiterative technique also requires an initial guess of the electricalproperties of each region, then uses a forward calculation of theexpected currents at the boundary to check the validity of the guess,iterating this process until the guess produces boundary currents closeto the measured values. The scale factor technique creates a “look up”table or neural net algorithm that allows one to correlate electricalproperties or the interior of the sample with externally measuredparameters using a large data set of model calculations. Because oflimitations of the model and the need to extrapolate results to keep thesize of the data sets reasonable, the scale factor technique has limitedaccuracy, but it does not require prior knowledge of approximate sampleelectrical properties. In fact, the results of the scale factorcomputation may serve as an initial estimate for the iterativetechnique. Both techniques are computationally intensive.

SUMMARY OF INVENTION

[0008] The present invention solves the problems associated with priorelectrical parameter imaging techniques by providing an apparatus andmethod that generates an accurate image of the electrical properties ofan object. More particularly, a charge correlation matrix is stored andemployed during image reconstruction to directly calculate the internalcharge distribution in the object from acquired surface chargemeasurements made on the object. From the calculated internal chargedistribution, images of internal electrical properties such asconductivity and dielectric constants may be produced.

[0009] A general object of this invention is to produce high resolutionimages of internal electrical properties without the need for a separateimaging system. Charge correlation matrices are pre-calculated andstored in the apparatus for use with selected sensor array geometriesand with prescribed image sizes and resolutions. The calculation of theinternal charge distribution is a straight forward multiplication of theacquired surface charge data by the appropriate, stored chargecorrelation matrix. Electrical properties images are easily producedfrom the resulting internal charge image.

BRIEF DESCRIPTION OF DRAWINGS

[0010]FIG. 1 is a block diagram showing an exemplary computer systemuseful for implementing the present invention;

[0011]FIG. 2 is a planar view of a closed volume in space;

[0012]FIG. 3 is a planar view of a closed volume in space showing therelationship between the measured exterior total charges Q_(j) and theinterior total charges qi;

[0013]FIG. 4 is a planar view of a closed volume in space being measuredby a conventional electrical property imaging technique;

[0014]FIG. 5 is a three dimensional representation of a sample holderwhich is employed when practicing the present invention;

[0015]FIG. 6 is a block diagram showing an apparatus which is employedwhen practicing the present invention;

[0016]FIG. 7 is a block diagram showing a controller unit of theapparatus of the present invention;

[0017]FIG. 8 is a control flow diagram showing the operation and methodof the apparatus;

[0018]FIG. 9 is a control flow diagram showing the production of anempty data set for use in the process of FIG. 8;

[0019]FIG. 10 is a control flow diagram showing the generation of finalmeasurements of electrical properties for use in the process of FIG. 8using the preferred embodiment of the present invention; and

[0020]FIG. 11 is a pictorial representation of a diagonal matrix.

DESCRIPTION OF THE PREFERRED EMBODIMENT

[0021] The preferred embodiment of the invention is implemented in aprogrammed computer shown in FIG. 1. The computer system 102 includesone or more processors, such as a processor 104, connected to acommunication bus 106. The computer system 102 also includes a mainmemory 108, preferably random access memory (RAM), and a secondarymemory 110. The secondary memory 110 includes, for example, a hard diskdrive 112 and/or a removable storage drive 114, representing a floppydisk drive, a magnetic tape drive, a compact disk drive, a programcartridge and cartridge interface (such as that found in video gamedevices), a removable memory chip (such as EPROM, or PROM), etc. whichis read by and written to by a removable storage unit 116. The removablestorage unit 116, also called a program storage device or a computerprogram product, represents a floppy disk, magnetic tape, compact disk,etc. As will be appreciated, the removable storage unit 116 includes acomputer usable storage medium having stored therein computer softwareand/or data. The removable storage drive 114 reads from and/or writes toremovable storage unit 116 in a well known manner.

[0022] The computer system 102 may also include other similar means forallowing computer programs or other instructions to be loaded. Suchmeans can include, for example, a communications interface 118 whichallows software and data to be transferred between computer system 102and external devices. Examples of communications interface 118 include amodem, a network interface (such as an Ethernet card), a communicationsport, etc. Software and data transferred via communications interface118 are in the form of signals which can be electronic, electromagnetic,optical or other signals capable of being received by communicationsinterface 118.

[0023] In this document, the term “computer program product” refers toremovable storage unit 116, a hard disk installed in hard disk drive112, and signals transferred via communications interface 118. Thesecomputer program products are means for providing software to a computersystem 102. In the preferred embodiment where the invention isimplemented using software, the software may be stored in main memory108, or in a computer program product and loaded into computer system102 using removable storage drive 114, hard disk drive 112, orcommunications interface 118. Data structures such as the chargecorrelation matrices discussed below may be loaded and stored in thesame manner. The software, when executed by the processor 104, causesthe processor 104 to perform the functions of the invention as describedherein. In an alternative embodiment, the invention may be implementedprimarily in hardware using, for example, a hardwired state machine.Implementation of the state machine so as to perform the functionsdescribed herein will be apparent to persons skilled in the relevantarts.

[0024] The computer system 102 also includes an output peripheral device122 and one or more input devices 120. The output peripheral device maybe a computer screen or monitor on which a graphical user interface,including a windows environment, may be displayed. The input devices 120include, for example, a keyboard, a mouse, a light pen, apressure-sensitive screen, etc., which provide a user with thecapability of entering input to the computer system 102.

[0025] The underlying mathematical theory of the imaging technique ofthe present invention will now be described with reference to FIGS. 2-4.FIG. 2 is a planar view of a closed volume space 200 surrounded by asurface 202 that contains a sample 204 and an interior region F 206,such that region F 206 is the space between the sample 204 and thesurface 202. The sample 204 comprises a plurality of connectedsubregions which for convenience are labeled: subregion A 208, subregionB 210, subregion C 212, subregion D 214, and subregion E 216. Eachsubregion 208-216 may be composed of a different material, such asdifferent tissues in a human subject.

[0026] When an electromagnetic field at some specified frequency (f) isapplied to the sample 204 in the closed volume space 200, a total chargeis produced only where the electrical properties change, such as at theboundaries between each subregion 208-216 of the sample 204 where thereis a dissimilarity in the dielectric constant and conductivityelectrical properties of each subregion 208-216. These total chargeswill in turn induce a redistribution of the total charges on the surfaceof the closed volume space 200. It is assumed that these induced chargedistributions result from both free charges (free to move individually)as well as polarization charges located on the surface 202 of the closedvolume space 200. The charges on the surface 202 are also total (freeplus polarization) charges wherein the total charge on a point on thesurface 202 is indicated with a capital “Q”, while the total charge on apoint in the interior of the closed volume space 200 is indicated with asmall “q.” It is important to note that the measurement of the totalcharge Q can involve either an actual measurement of the charge Q or thecharge Q as derived from a small increment of the electrical current, I,which is the rate of change of the charge Q with time.

[0027] The total charge Q at a point on the surface 202, and the totalcharge q at a point in the interior can be connected via electromagnetictheory. The fundamental theorem of electrostatics shows that an interiortotal charge q and a total charge Q on the surface 202 are uniquelyrelated. However, when time varying electric fields are applied toelectrical media they induce currents in the media. These currents inturn produce time varying magnetic fields that can induce electricfields in addition to the applied electric field via Faraday's law. Thisextra contribution to the electric field is negligible at lowfrequencies and can be ignored. We will use this so-called“quasi-static” approximation.

[0028]FIG. 3 is a planar view of the closed volume space 200 showing therelationship between the total charge Q at a point on the surface 202and a total charge q at a point in the interior that are connected viaThe Greens Function. Specifically, The Greens Function connects a totalcharge Q on the surface 202 at point j with an interior total charge qat point

q_(i)

Q_(j)

[0029] This relationship provides the desired information about theelectrical properties of the interior subregions 208-216 of sample 204.FIG. 3 illustrates the coordinate system and some of the relevantgeometry used in this discussion. The notation used in the coordinatesystem for the field point 304, the source point 306 and surface point302 are {overscore (X)}, {overscore (X)} prime ({overscore (X)}′) and{overscore (X)} double prime ({overscore (X)}″) respectively. Byassociating the total charges q inside the sample 204 at the sourcepoint 306 with the total charges Q at the surface point 302, an enhancedimage of the interior of the sample 204 can be generated. Therefore, theposition at which the electric field is measured is field point 304.

[0030] The imaging technique of the present method differs significantlyfrom the conventional electrical property imaging techniques. FIG. 4 isa planar view of a closed volume space 200 being measured by suchconventional imaging techniques. A network of lumped circuit elementsrepresents the electrical properties of the sample. With such a method,currents are injected at known places, e.g., P1 402, on the surface 202of the closed volume space 200 and extracted at known places, e.g., P2404. The voltages on the surrounding sensors are then measured and thelumped circuit impedances are computed from the set of current-voltagemeasurements. In contrast, the technique of the present invention allowsone to fully describe the wave-like nature of the electric fields in theclosed volume space 200 and the measuring volume and does not requireany specific assumption regarding the structure of a lumped circuitelement network or of the equivalent circuits used to characterize thesubregions 208-216 of the sample 204 being measured.

[0031] Applying the Maxwell Equations of electromagnetic theory to theproblem as just described results in Equation 1A: $\begin{matrix}{{\left. {{\nabla{\cdot \left. \left\lbrack {\sigma + {{j\omega ɛ}_{0}ɛ_{r}}} \right. \right)}}\left( {- {\nabla\Phi}} \right)} \right\rbrack = 0}\text{Where:}{\sigma = {{conductivity}\quad {constant}}}{ɛ_{r} = {{relative}\quad {dielectric}}}\text{}{ɛ_{0} = {{dielectric}\quad {constant}\quad {of}\quad {free}\quad {space}}}\text{}{\Phi = {{potential}.}}} & \text{(1A)}\end{matrix}$

[0032] In addition, a standard result of electromagnetic theory is theconnection between the potential, (Φ)), and the total charge density, ρ,known as the Poisson Equation, Equation 1 B: $\begin{matrix}{{\nabla^{2}\Phi} = \frac{\rho_{Total}}{ɛ_{0}}} & \text{(1B)}\end{matrix}$

[0033] where P_(Total) is the volume total charge density. The electricfield E is obtained from the following equation:

{overscore (E)}=−∇Φ  (1C)

[0034] The Equations 1A and 1B show that for determining the electricfield E and the scalar potential phi (Φ), the charge densities that areimportant are related to the total charge, i.e., the free charge pluspolarization charge.

[0035] Other methods for imaging the electrical properties attempt tocompute the dielectric constant and conductivity of each region directlyfrom the measurements. In the current invention we compute the totalinternal charges as an intermediate step. One advantage of seeking thecharges rather than going directly for the conductivity or dielectricconstant is that the internal charges, which totally govern theelectrical picture, appear essentially only at boundaries that exist atdiscontinuities within the object, thus there are far fewer values tocompute. Equation 2 below shows this since the gradient of theconductivity and the gradient of the dielectric constant contribute tothe total charge density. Therefore, total charge depends on the ratewith which the conductivity and the dielectric constant change withdistance. $\begin{matrix}{\rho_{Total} = \frac{\left\lbrack {{\nabla\sigma} + {{j\omega}{\nabla\left( {ɛ_{0}ɛ_{r}} \right)}}} \right\rbrack \cdot {\nabla\Phi}}{\sigma + {{j\omega ɛ}_{0}ɛ_{r}}}} & (2)\end{matrix}$

[0036] A standard theorem in electromagnetic theory is the UniquenessTheorem. The Uniqueness Theorem for the quasistatic case states that ifthe potential or its normal derivative is known on a surface surroundinga closed volume, then the potential at a field point 304 can be uniquelydetermined. It is important to note that either the potential or thenormal derivative of the potential need be known but not both. In fact,the problem would be over determined if both were known. While it ispossible to define the problem with the potential known on some portionof the bounding surface and the normal derivative on other portions,Equation (3) below considers the simple case where the potential on thesurface 202 is known. This is known as the Dirichlet boundary condition.

[0037] Equation 3 is the solution to Poisson's Equation (Equation 2)using the Green's Function. $\begin{matrix}\begin{matrix}{{\Phi \left( \overset{\_}{X} \right)} = \quad {{\frac{1}{4{\pi ɛ}_{0}}{\int{{{\rho_{Total}\left( \overset{\_}{X^{\prime}} \right)} \cdot {G_{D}\left( {\overset{\_}{X},\overset{\_}{X^{\prime}}} \right)}}{\tau}}}} -}} \\{\quad {\frac{1}{4\pi}{\oint{{\Phi \left( \overset{\_}{X^{\prime}} \right)}\frac{\partial{G_{D}\left( {\overset{\_}{X},\overset{\_}{X^{\prime}}} \right)}}{\partial n^{\prime}}{S}}}}}\end{matrix} & (3)\end{matrix}$

[0038] Where G_(D) is the Dirichiet Green's Function, dτ is an elementof volume and S is an element of surface surrounding the volume τ. Thefirst term in this equation is an integral over the entire volume andthe second term is an integral over the surface of that volume.

[0039] Equation 3 is the potential at the field point 304 as determinedby the total charge q on the interior and the potential on the surface202, exactly as the Uniqueness Theorem predicts. The solution isobtained in terms of a geometrical function, the Green's Function, whichis a standard treatment. When a sample 204 is present, both the volumeintegral of the total charge density ρ_(Total) and the surface integralof the potential Φ are present. If the same potential distribution onthe surface is considered but with no sample present, then the chargedensity term is zero but the surface integral term remains the same. Thesurface term (the second integral in Equation 3) is unchanged byinserting the sample 204 because the voltage is set to pre-determinedvalues on the surface 202 and kept at those values before and afterinserting the sample 204. Because of this, when the two terms aresubtracted, the remaining expression involves only the Green's Function(which is a known quantity for a given shape of the array of measuringsensors) and the charge density. Therefore, it is convenient to use thedifference in the potential between the case when a sample 204 isinserted and when a sample 204 is not inserted between the sensors. Thispotential difference can be related to the charges at the surface 202 bytaking the normal derivative of the potential difference, which producesthe normal component of the electric field since By Gauss's law, thenormal component of the field near a conducting surface is directlyproportional to the charge per area on that surface.

[0040] The above continuum model of the relationship of surface charge Qto the internal charge q can be expressed as the sum of measureddiscrete charges set forth below in equation (4). Equation (4) showsthat those charges Q_(j) at the surface 202 labeled by the index “j”will be related to the charges q_(i) on the interior labeled by theindex “i” by a matrix element involving both “j” and “i” wherein theconnecting matrix element is simply the normal derivative of the Green'sFunction: $\begin{matrix}{{\Delta \quad Q_{jTotal}} = {{{Q(j)}_{Total}^{Full} - {Q(j)}_{Total}^{Empty}} = {\sum\limits_{i}{\frac{\partial{G_{D}\left( {j,i} \right)}}{\partial n_{j}^{''}} \cdot q_{i}}}}} & (4)\end{matrix}$

[0041] Equation 5 below shows that this series of equations in “j” canbe written down and grouped together in matrix formulation involving acharge on the surface 202 as a vector, with each term of the vector oneof the total charges. For the charges on the surface 202, a capital “Q”is used and they are related to a similar vector for which each term isone of the total charges on the interior using the small “q.”$\begin{matrix}{\overset{\_}{\Delta \quad Q} = {\frac{\overset{\_}{\overset{\_}{\partial G_{D}}}}{\partial n^{''}} \cdot \overset{\_}{q}}} & (5)\end{matrix}$

[0042]1This series of equations is inverted to give the charges on theinterior, “q”, provided that the matrix itself has an inverse. Morespecifically, if an inverse exists, when the Green's Function derivativematrix is multiplied by its inverse, a unit matrix results as shown inequation (6). $\begin{matrix}{{\frac{\overset{\_}{\overset{\_}{\partial G_{D}}}}{\partial n} \cdot \overset{\_}{\overset{\_}{\left( \frac{\partial G_{D}}{\partial n} \right)^{- 1}}}} = {I = \begin{pmatrix}1 & 0 & \cdots & \cdots & 0 \\0 & 1 & 0 & \cdots & \cdots \\\cdots & 0 & 1 & 0 & \cdots \\\cdots & \cdots & \cdots & \cdots & \cdots \\0 & \cdots & 0 & 0 & 1\end{pmatrix}}} & (6)\end{matrix}$

[0043]FIG. 5 is a three dimensional representation of a sample holder500 having a sample 204 contained therein. A plurality of sensors 502are arrayed on each side of the sample holder 500 in a matrixconfiguration. This three-dimensional implementation of the invention ispreferred, however, in the following discussion a two dimensionalimplementation is described. We will continue further development of thetheory for this imaging method using the sensors in a two dimensionalcross-section through the sample holder 500. This is shown best in FIG.6. The resulting 2D sample holder apparatus 600 includes two opposingrows of sensors spaced apart by a distance b and extending along alength a, and two opposing rows of sensors spaced apart by distance band extending along length a. The preferred embodiment of the presentinvention is described in terms of a rectangular sample holder. It wouldbe readily apparent to one of ordinary skill in the art to use a sampleholder of a different geometric cross-sectional shape. However, whensuch a new shape is used, the sine function used to generate the Fouriertransform values (to be described below) would be replaced by a functionchosen from an appropriate orthonormal set of functions particular tothe new shape of the sample holder.

[0044] A sample 204 is placed inside the sample holder 500 with amatching medium 504 that matches the impedance between the sample 204and the sensors 502. More specifically, the matching medium 504 matchesthe average impedance of the entire sample 204. The sensors 502 are anysensing element that can be probed to determine the electric chargedeposited on the sensor. Each sensor 502 serves the dual purpose ofproviding a controlled voltage to the sample 204, thereby generating theelectromagnetic field, and of measuring the resulting electricalcurrent, or total charge Q, over a fixed time interval.

[0045] It is possible to acquire sufficient charge data to reconstructan image with a single orientation of the subject and applied electricfield E. However, the signal to noise ratio (SNR) can be significantlyimproved if multiple measurements are made. By applying a sinusoidalpotential of constant amplitude Al across the top 510 of the rectangularspace 500, and a similar potential of a different amplitude A2 on thebottom 512 of the rectangular space 500, with similar potentials havingan amplitude varying continuously from Al to A2 on equivalent portionsof surfaces 506 and 508, a uniform electric field E is produced in theobject in the vertical direction. Similarly a uniform electric field Eis produced in the horizontal direction by reversing the roles ofsurfaces 510 and 512 with 506 and 508. The object can also be rotatedwith respect to the sample holder 500 to provide four possiblecombinations of sample 204 direction and electrical field direction.These four possible measurement conditions are set forth in TABLE 1.TABLE 1 Orientation (OR) Sample Direction (S) Electrical Field Direction(E) 1 ↑S ↑E 2 ↑S →E 3 →S ↑E 4 →S →E

[0046] Sample 204 orientations can be changed by rotating the plates ofthe sample holder or by rotating the sample 204 itself. Such rotationcan be achieved by hand or by an automated means. From these fourorientations, fifteen combinations of the resulting acquired data can bemade. These fifteen possible combinations of field direction and sampleorientation are set forth in TABLE 2. TABLE 2 Combination Orientations 11 2 2 3 3 4 4 5 1,2 6 1,3 7 1,4 8 2,3 9 2,4 10 3,4 11 1,2,3 12 1,2,4 131,3,4 14 2,3,4 15 1,2,3,4

[0047] The total charge Q_(j) on the surface 202 is measured bymeasuring the current as a function of time, I(t), setting the charge oneach sensor equal to zero at time t=0 and computing the total charge byintegrating the current over time from t=0 to the desired time t. Forexample, when the current is determined from the phase sensitivedetector measurements to have an amplitude A and phase α for a sine wavetime dependence (I_(j)(t)=Asin(ωt+α)) the total charge is given by$\begin{matrix}{{Q_{j}(t)} = {{\int_{0}^{t}{{I(t)}{t}}} = {{Q_{j}(0)} - {\left( {1/\omega}\quad \right)A\quad {\cos \left( {{\omega \quad t} + \alpha} \right)}}}}} & \text{(6A)}\end{matrix}$

[0048] Equation 7 below shows how to get rid of the double sum that isgoing to appear in the expression for the total charge Q_(j) on thesurface 202 of a sample 204. Once the Green's Function is expanded in acomplete set of orthogonal functions (which is just the sine function),the result is a sum over the parameter “L” which appears inside the sinefunction in Equation 7 and also a sum over the charges qi which appearin Equation 4. Multiplying by the appropriate sine function for givenvalue “L”, and summing up over the top side 510 or the bottom side 512,results in the elimination of the sum over “L”, thereby leaving just oneterm remaining. This result occurs because of the orthogonality propertyof sine and cosine functions. The accuracy can be further improved byadding the results from the top 510 and bottom 512, resulting in theequation for a given value of “L” for the Fourier Transform (the sinetransform) as shown in Equation 8. $\begin{matrix}{{{FT}(L)} = {\int_{0}^{a}{{\left\lbrack {{\Delta \quad {Q_{Top}(x)}} - {\Delta \quad {Q_{Bottom}(x)}}} \right\rbrack \cdot {\sin \left( \frac{L \cdot \pi \cdot x}{a} \right)}}{x}}}} & (7) \\{{{FT}(L)} = {\frac{{\Delta \quad S}}{\pi^{2}}{\sum{\frac{{\sin \left( \frac{L \cdot \pi \cdot x_{i}^{\prime}}{a} \right)} \cdot {\cosh\left( \frac{L \cdot \pi \cdot \left( {\frac{b}{2} - y_{i}^{\prime}} \right)}{a} \right)}}{\cosh \left( \frac{L \cdot \pi \cdot b}{2a} \right)} \cdot q_{i}}}}} & (8)\end{matrix}$

[0049] For each value of “L”, one equation can be produced each of whichinvolves the sum over the charges labeled by “i,” the matrix elements ofwhich are shown in Equation 8. This set of equations becomes somewhatbetter conditioned, i.e., one equation differs more from the other, ifthe length of the small side “b” is small compared to two times thelength of the long side “a.” Nonetheless, highly accurate results can beobtained for a=b.

[0050] We have discovered that it is possible to actually solve theseequations (8) rigorously and produce an exact minimum noise solution tothe problem, and thus to produce an accurate representation of thecharges q in the interior of the object. Once this has been done, theinterior charge distribution image can be used to build the solution forthe potential everywhere on the interior of the object using the knownGreen's Function Solution presented in Equation (3) above. Once thepotential everywhere on the interior of the object is known, theelectrical fields can be easily generated from those potentials usingEquation (1C). One can then obtain the change in the electrical field asyou go from one point in the interior to another, which then produces anestimate of the electrical properties at every point in the interior ofthe object.

[0051] Equation (8) is an expression for the electric charge if one addsthe Fourier transforms for the surface charges measured along a row ofdetector elements disposed on opposite sides of the object. Thecoordinates x′ and y′ are the location of the internal charge q, wherex′ is along the direction of the row of opposing detector elements andy′ extends perpendicular to and between the two rows of opposingdetector elements. The functional dependence on the position of theinterior charge q in the direction parallel to the measuring plates (x′)is a trigonometric sine function. Each sine function is indexed by aninteger L, where L can vary from one to an arbitrarily high number n. Inthis preferred embodiment n is set equal to twice the number of pixelsdesired in the prescribed image (i.e., twice the number of interiorcharge points q_(i)).

[0052] We also note that expressing these equations in vector matrixnotation produces a matrix (which we call M (L,i)) where each row of thematrix is perpendicular to every other row because of the orthogonalityproperty of the sine functions. $\begin{matrix}{{{{FT}(L)} = {\sum\limits_{i}{{M\left( {L,i} \right)} \cdot q_{i}}}}\text{Where:}{{M\left( {L,i} \right)} = {\frac{{\Delta \quad S}}{\pi^{2}} \cdot \frac{{\sin \left( \frac{L \cdot \pi \cdot x_{i}^{\prime}}{a} \right)} \cdot {\cosh\left( \frac{L \cdot \pi \cdot \left( {\frac{b}{2} - y_{i}^{\prime}} \right)}{a} \right)}}{\cosh \left( \frac{L \cdot \pi \cdot b}{2a} \right)}}}} & (9)\end{matrix}$

[0053] It is possible using standard theorems of matrix theory to showthat the columns of the matrix are also orthogonal provided that therows are “normalized”. (To normalize, take the vector dot product ofeach row with itself, take the square root of the number that's producedand divide each element in the row by the square root.) But if thecolumns of the matrix are orthogonal, then the entire matrix is an“orthogonal matrix”. The inverse of the matrix is just its transpose(where the rows and the columns are interchanged) and this means thatthe desired inverse of equations (8) are a straightforward calculation.

[0054] To test this, we normalized the rows of the matrix {doubleoverscore (M)} and we then further took the matrix dot product of thetranspose of that matrix {double overscore (M)}^(T) with the original.$\begin{matrix}{{{\overset{\overset{\_}{\_}}{M}}^{T} \cdot \overset{\overset{\_}{\_}}{M}} = \overset{\overset{\_}{\_}}{D}} & (10)\end{matrix}$

[0055] The results are shown in FIG. 11. The vertical axis representsthe value of the matrix {double overscore (D)} element, and the twohorizontal directions represent the row and the column number from whichthat element is taken. We note that the resulting matrix {doubleoverscore (D)} is substantially diagonal; i.e. the non-zero elements arealong the diagonal of the matrix. When matrix {double overscore (D)} ismultiplied by the vector representing the internal charges ({overscore(q)}), only one of those charges is picked out for multiplication ofeach row. $\begin{matrix}{{{{\overset{\overset{\_}{\_}}{M}}^{T} \cdot \overset{\_}{F\quad T}} = {{{\overset{\overset{\_}{\_}}{M}}^{T}{\overset{\overset{\_}{\_}}{M} \cdot \overset{\_}{q}}} = {{\overset{\overset{\_}{\_}}{D} \cdot \overset{\_}{q}} = {\begin{bmatrix}D_{11} & 0 & 0 & \cdots & 0 \\0 & D_{22} & \quad & \quad & 0 \\0 & \quad & D_{33} & \quad & \vdots \\\vdots & \quad & \quad & \ddots & 0 \\0 & 0 & \cdots & 0 & D_{i\quad i}\end{bmatrix} \cdot \overset{\_}{q}}}}}{{{so}\quad {that}\quad q_{i}} = {\sum\limits_{L}{{{M^{T}\left( {L,i} \right)} \cdot F}\quad {{T(L)}/D_{i\quad i}}}}}} & (11)\end{matrix}$

[0056] Algebraically, equation (11) is an expression for each separateinternal charge q_(i). All of the factors needed for this computationcan either be measured or computed, and thus each element of chargeq_(i) in the interior of the object can be calculated separately. Thematrix {double overscore (M)}^(T) is referred to herein as a “chargecorrelation matrix”, and by calculating and storing the elements of thischarge correlation matrix prior to image acquisitions, thereconstruction of electrical property images from acquired surfacecharge data is a relatively simple procedure.

[0057] An advantage of the present invention is that the chargecorrelation matrix can be determined for a given set of measuring platesand image resolution, and stored in the computer ahead of time. It iscontemplated that a plurality of charge correlation matrices will bestored in system memory. One of these matrices will be selected based onthe particular sample holder configuration used and the particular imageprescription chosen. To evaluate the charges on the interior, therefore,we need only measure and produce the Fourier transforms of the measuredsurface charges Q. The process of inverting the data to produce theinterior charges q is merely a matter of multiplying the transformedcharge data by the selected charge correlation matrix. There is noiterative process and there is no need to make an initial guess at themagnitude of the electrical properties of each structure in theinterior. Most importantly, it is also not necessary to make ameasurement of the shape of the interior structures using a separateimaging system.

[0058] The final steps in taking the interior measured charges andproducing the potentials and/or electrical fields on the interior of theobject can be accomplished by inserting the charges q into the Greens'functions solution Equation (3). We note that there are two terms in theexpression for the interior electrostatic potentials: A) a volumeintegral involving the charges q that were just calculated above; and B)an integral over the surface involving known potentials that are set atthe surface. The first integral is obtained just using the chargescalculated above. The second integral is obtained just as easily,because the potentials that are set at the surface sensors aredetermined by the experimenter and are known. Therefore everything isknow and can be calculated by a simple plug-in operation and knownformula to get the interior potentials according to equation (3).

[0059] Once the interior potentials are known, the electric fieldeverywhere can be obtained from the rate of change of the potential ineach direction as given in Equation (1C). Once the electric fields areknown everywhere in the interior the electrical properties (ε_(r) and σ)of each region within this object can be computed from the change of thenormal component of this electric field across each boundary within theobject. Therefore, by starting with the known electrical properties ofthe medium surrounding the object, the electrical properties of adjacentregions in the object can be calculated. At each boundary of region(k)to region (k+1) the ratio of the normal components of the electricfields are related to the electrical properties in regions (k) and (k+1)as follows: $\begin{matrix}{\frac{{\overset{\_}{E}(k)}_{normal}}{{\overset{\_}{E}\left( {k + 1} \right)}_{normal}} = \frac{\left( {{\sigma \left( {k + 1} \right)} + {j\quad \omega \quad ɛ_{0}{ɛ_{r}\left( {k + 1} \right)}}} \right)}{\left( {{\sigma (k)} + {j\quad \omega \quad ɛ_{0}{ɛ_{r}(k)}}} \right)}} & (12)\end{matrix}$

[0060] Since the values of σ and ε_(r) are known for the matching medium(the region with k=1), use of Equation (12) will yield correspondingvalues for the next region (k=2). Applying the equation again for theboundary between region 2 and 3, we get the values for region 3, and soon until we have the electrical properties of the entire object.

[0061]FIG. 6 is a block diagram showing an apparatus 600 which isemployed to practice the present invention with a sample 204. In thepreferred embodiment, the apparatus 600 comprises a capacitive sensorarray 604 a-d on each side of the sample 204 for detecting the pointtotal charges Q at each point on the surface 202 of the sample 204.Specifically, there is a top array 604 a, right side array 604 b, bottomarray 604 c, and left side array 604 d, which collectively form a sampleholder 500. Therefore, the surface 202 of the sample 204 is the surfaceof the capacitive sensor arrays 604 a-d (the sample holder 500). Whenthe sample 204 is thick, the preferred embodiment comprises at least twocapacitive sensor arrays 604 a, c. However, it would be readily apparentto one of ordinary skill in the art to use an apparatus and method ofthe present invention using one or more such capacitive sensor arrays604 for detecting the total charges Q at the surface 202 of a sample204.

[0062] In operation, which is described in greater detail below, asample 204 is placed between the capacitive sensor arrays 604 a-d withtwo sides of the sample 204 being reasonably close to the capacitivesensor arrays 604 a-d and with a matching medium 602 applied around thesample 204 as a buffer between the sample 204 and the capacitive sensorarrays 604 a-d. Enough matching medium 602 is used to fill up theremainder of the sample holder 500 surrounding the sample 204. Thispreferred embodiment of matching medium 602 minimizes error in readingthe total charges of the sample 204. A matching medium 602 is used toproduce appropriate total charges Q at the segmented sensor plate 606(which is equal to the surface 202 of the sample 204). Specifically, amatching medium 602 is chosen for a sample 204 such that the medium 602optimizes the accuracy of the resulting dielectric constant andconductivity of each subregion 208-216 of the sample 204.

[0063] Each capacitive sensor array 604 a-d comprises an identicalstructure. For the purpose of convenience only, one capacitive sensorarray 604a is described, however, the description is equally applicableto the other capacitive sensor arrays 604 b-d of the present invention.A capacitive sensor array 604 a comprises a segmented sensor plate 606having a plurality of controller units 608 a-f applied thereon andarranged in a matrix format. Each controller unit 608 a-f applies thedesired voltage to the sample 204 at a desired frequency, therebygenerating an electromagnetic field, as well as detecting the totalcharge Q at the segmented sensor plate 606. A controller unit 608 a isdescribed below in greater detail.

[0064] A power supply 610, frequency source 612, and a computer system614 are connected to the capacitive sensor arrays 604 a-d by a bus 616.The power supply 610 provides the necessary DC power for the componentsof the sensor array 604 and the frequency source 612 provides thenecessary alternating voltage to the capacitive sensor arrays 604 a-dfor measuring the sample 204 contained therein. In the preferredembodiment, the frequency source 612 provides both a zero degreecomponent 618 (in-phase) and a ninety degree component 620 (quadraturephase) of the alternating voltage being output from the controller units608 a. Further, the frequency can be changed so that the apparatus 600can make multiple frequency measurements as needed. When imagingtissues, for example, frequencies ranging from 100 Hz to 100 MHz may beemployed to exploit the differences in electrical characteristicsbetween various tissue types. The computer system 614 controls thecomponents of the apparatus 600 and performs the calculations needed togenerate the dielectric constant and conductivity electrical propertiesof the sample 204.

[0065]FIG. 7 is a block diagram illustrating the structure of acontroller unit 608 a of a capacitive sensor array 604 a. Forconvenience purpose only, one controller unit 608 a is described,however, the description is equally applicable to the other controllerunits 608 b-f. In the preferred embodiment, controller unit 608 acomprises multiple components, including but not limited to, a localcontroller 702, an A/D converter 704, a multiplexer 706, a firstSample/Hold (S/H) component 708, a second S/H component 710, anout-of-phase phase (90 degree)phase sensitive detector 712, an in-phasephase (0 degree)phase sensitive detector 714, an auto-gain controller716, a segment voltage controller 718, and a pure resistor 724. Thesecomponents communicate via a local digital bus 722. Further, thesecomponents communicate with the other parts of the apparatus 600 via thebus 616. More specifically, the controller unit 608 a transmits andreceives data to/from the computer system 614, receives power 724 fromthe power supply 610, and receives in-phase and out-of-phase alternatingvoltage from the frequency source 612 over the bus 616. In operation,the computer system 614 commands the segment voltage controller 718 of25 the controller unit 608 a to set a given alternating voltage at aspecified frequency. The pure resistor 724 is used to create a voltagethat is an accurate measure of the magnitude and phase of the electricalcurrent. Also, the computer system 614 engages the auto-gain controller716 to maximize the voltage on pure resistor 724 reflective of theelectrical current, or charge, inputted to the phase sensitive detectors712, 714. Next, the computer system 614 commands the in-phase phasesensor detector 712 and the out-of-phase phase sensor detector 714 toread the total charges Q on the segmented sensor plate 606. Thecontroller unit 606 transmits these total charges Q to the computersystem 614 via the first S/H 708, the second S/H 710, the multiplexer706, the A/D converter 704 and the local controller 702. Thesecomponents operate using conventional methods. It should be readilyapparent to one of ordinary skill in the relevant arts to convert theanalog format of the total charges Q to digital values and then transmitthese digitized total charges Q to the local controller 702 and thecomputer system 614 for further processing.

[0066] While we have described an embodiment using phase sensitivedetectors to determine the amplitude and phase of the current at eachsensor, it will be readily apparent to one skilled in the art that thereare alternate methods to perform this task. For example, the currentdata could be digitized and the resulting digital signal could then beanalyzed with a Fourier Transform Analyzer. This alternate method wouldbe useful for creating images at several frequencies of the appliedfield simultaneously, thus saving time compared to the use of phasesensitive detectors, where the frequency would have to be changedsequentially for each sensor if only one set of quadrature phasesensitive detectors were to be employed for each sensor.

[0067] The apparatus 600 may be incorporated into a standard tomographicdevice, e.g., an x-ray CT or MRI system. This combination allows theimaging system to produce a high resolution image that may be registeredwith the image produced by the apparatus 600. However, it is anadvantage of the present invention that high resolution images of theelectrical properties of the subject can be produced without input froma separate imaging system.

[0068] FIGS. 8-10 are flow charts illustrating the operational controlof the apparatus 600. Referring to these figures, the method of usingthe apparatus 600 with a standard imaging system 620 is described belowin detail. It can be appreciated, however, that this apparatus 600 canalso be operated in a stand-alone mode. Control starts at step 802 atwhich the computer system 614 inputs initial start-up data, includingbut not limited to, a source indicator for empty data values, and theorientation of the sample 204 in the sample holder 500. As indicatedabove, a plurality of precalculated charge correlation matrices arestored in memory and as part of the initialization process, one of theseis selected. This selection is made based on the particular imageacquisition parameter prescribed by the user, such as image size andimage resolution. At this step, the computer system 614 also generatesan electromagnetic field within the sample holder 500 of the apparatus600 according to a predetermined alternating voltage of a selectfrequency.

[0069] After start-up, processing proceeds to step 804. In step 804, thecomputer system 614 of the apparatus 600 sets the electromagnetic fieldaccording to a given voltage and frequency of each controller unit 606and sets the orientation of the sample 204. In the next step 806, thecomputer system 614 sets the address pointer for the empty data values.The “empty” data values are the total surface charges Q when a sample204 is not in the apparatus 600; that is, only the matching medium 602is present in the sample holder 500. Step 806 is described in greaterdetail below.

[0070] Processing then continues to step 808. In step 808, an object 204to be imaged is placed between the capacitive sensor arrays 604 a-d, orthe sample holder, of the apparatus 600. Proceeding to step 810, theapparatus 600 waits for a trigger from the imaging system to which it isconnected. By using this triggering mechanism, the sensor measurementsof the total surface charge Q and the acquired image data are acquiredat precisely the same time, for the same condition of the object 204.The imaging system may be used for triggering the apparatus 600 at aconvenient time for acquiring optimum information about the object 204,such as at specifically monitored portions of the cardiac or pulmonarycycle of a human subject. Upon receiving the trigger from the imagingsystem, the apparatus 600 continues to step 812.

[0071] In step 812, the computer system 614 commands the controllerunits 606 a-f to poll their sensors, the phase sensitive detectors 712,714, and measure the total charge Q on the segmented sensor plate 606.This represents the full data values which are the total surface chargesQ when the sample 204 is located within the sample holder 500 of theapparatus 600. This polling of the sensors is performed in parallel sothat the time averaging for any one measurement is the same as the timeaveraging for all measurements. Once the data is received, the apparatus600 continues to step 814. In step 814, the computer system 614 sets adata pointer to the full data it just read from the polling of thesensors and proceeds to step 816.

[0072] In step 816, the computer system 614 determines whether there areadditional measurements to be made with a new orientation for the sample204 or the electric field as it is contained within the sample holder500 of the apparatus 600. If another measurement is to be made, thecomputer system 614 returns to step 804 to restart its data acquisitionsequence. After all the prescribed measurements are made, the computersystem 614 computes the electrical properties for each pixel in thesubject 204 and produces an image thereof. Step 818 is described ingreater detail below. When the electrical properties image has beenproduced, the computer system 614 proceeds to step 820, therebycompleting its processing.

[0073] The final measurements of dielectric constant and conductivityelectrical properties of a object 204 can be used in various ways. Inthe preferred embodiment, the final measurements are used to indicatewhether certain tissue in a sample 204 is living or dead, or whether thetissue has a pathology such as clogged arteries and veins, or acancerous tumor. This information may be presented to a user either in anumerical or enumerated format, a graphical format, or a combination ofboth. However, in the preferred embodiment the calculated electricalvalues are used to modulate the intensity and/or color of correspondingimage pixels.

[0074]FIG. 9 is a block diagram illustrating the processing of step 806which sets the data pointer for the empty data set. The process beginsat step 902 and immediately proceeds to step 904 which determines thesource of values to be used in the empty data set. If stored data valuesfrom a database are to be used, the computer system 614 proceeds to step906. In step 906, the computer system 614 sets a pointer to the storagelocation of the empty data set values according to the appropriatedatabase values and continues to step 914, thereby returning to FIG. 8and step 808.

[0075] Referring again to step 904, if the computer system 614determines that it must make measurements for the empty data set, thecomputer system 614 proceeds to step 908. In step 908, the computersystem 614 waits for a trigger from the imaging system. The imagingsystem may make a preliminary image of the empty sample holder 500 andtransmit this imaging data to the computer system 614 of the apparatus600, thereby triggering the apparatus 600 to continue its processing atstep 910. It is important to note that the imaging system need not makean image of an empty sample holder 500, but doing so allows for thechecking of the data.

[0076] In step 910, the computer system 614 commands the phase sensitivedetectors 712, 714 of the controller units 606 a-f to apply theprescribed voltages and poll their sensors and measure the total chargesQ on the segmented sensor plates 606. Once the charge data is received,the apparatus 600 stores this empty data set and continues to step 912.In step 912, the computer system 614 sets a data pointer to the addresslocation for the empty data set and proceeds to step 914, therebyreturning to FIG. 8 and step 808.

[0077] The electrical properties are calculated from the acquired chargedata in accordance with the steps shown in FIG. 10. As indicated at step1004, the first task is to subtract the empty data from the first set ofcharge measurements. This produces a set of surface charge data Q whichreflects the charges q inside the subject being imaged as set forthabove in equation (4). The resulting charge measurements Q acquired atcorresponding sensors 502 on opposite sides 506 and 508 or top andbottom 510 ad 512 of the sample holder 500 are either added together orsubtracted at step 1006. In the preferred embodiment they are added.

[0078] As indicated at step 1008, an array of L transformationcoefficients are then calculated from the resulting surface charge dataQ(x) as set forth in equation (7) above. In the preferred embodiment Lis set equal to the number of pixels in the prescribed image (i.e.number of internal charge locations q_(i)). The resulting Fouriertransform array FT(L) indicates values of the frequency constituents ofthe acquired surface charge data.

[0079] The next step as indicated at step 1010 is to multiply theFourier transform array FT(L) by the appropriate stored correlationmatrix{double overscore (M)}^(T). As indicated above, correlationmatrices are calculated for each sample holder 500 and each prescribableimage resolution. As indicated above by equation (11), this results in anumerical value for the charge q at each pixel i in the subject. Thesevalues are stored at step 1012 in an array to form a charge image.

[0080] As indicated at step 1014, a voltage image is calculated next asset for above in equation (3). This image indicates the voltage producedat every pixel in the image due to the applied external sinusoidalvoltage. As indicated at decision block 1016, a determination is thenmade if further measurements are to be made to increase the image SNR.As indicated above, these additional measurements can be made bychanging the direction of the field produced by the applied voltages andthe relative orientation of the subject as set forth above in TABLE 1.In the preferred embodiment all four orientations listed in TABLE 1 areacquired with the result that four voltage images are produced by theprocessing loop indicated at 1018.

[0081] The voltage images are combined to form a single, high SNRvoltage image at step 1020. This is done by computing the average valueat each image pixel. Other electrical parameters such as dielectricconstant and conductivity can now be calculated from the voltage image.As indicated at step 1022, this is done by first computing the electricfield everywhere using the reconstructed voltage image and Equation(1C). Then, the technique described above with regard to Equation (12)is performed in step 1024 to calculate the dielectric constant Ar andthe conductivity or at each image pixel. These calculated values may beused to produce separate images as described above, or the values may betabulated.

1. A method for producing an image of an electrical characteristic of anobject, the steps comprising: a) applying a voltage to the surface ofthe object with an array of sensor elements; b) measuring the surfacecharge at each sensor element that results from the applied voltage; c)transforming the surface charge measurements; d) calculating thedistribution of charges inside the object by multiplying the transformedsurface charge measurements by a stored charge correlation matrix; ande) producing an image of an electrical property of the object from theinternal charge distribution data calculated in step d).
 2. The methodas recited in claim 1 in which the array of sensor elements includes afirst set of sensor elements disposed on one side of the object and asecond set of sensor elements disposed on the other side of the object,and step a) is performed by applying the voltage across said first andsecond set of sensor elements.
 3. The method as recited in claim 2 inwhich the surface charge is measured at each sensor element by measuringthe current flowing therethrough over a preselected time interval. 4.The method as recited in claim 1 which includes inserting an impedancematching media between the sensor elements and the surface of theobject.
 5. The method as recited in claim 3 in which step b) includescombining the surface charge measurements from corresponding sensorelements in the first and second sets of sensor elements.
 6. The methodas recited in claim 5 in which the surface charge measurements arecombined by addition.
 7. The method as recited in claim 1 in which theimage produced in step d) indicates the electrical conductivitythroughout the object.
 8. The method as recited in claim 1 in which theimage produced in step e) indicates the dielectric constant throughoutthe object.
 9. The method as recited in claim 1 in which the imageproduced in step e) is produced by setting the intensity of each pixelin the image to a level corresponding to the value of the electricalproperty at the corresponding location in the object.
 10. The method asrecited in claim 1 in which the image produced in step e) is produced bysetting the color of each pixel in the image as a function of the valueof the electrical property at the corresponding location in the object.11. The method as recited in claim 1 in which the transformationperformed in step c) is a Fourier transformation.
 12. An imaging systemfor producing an image of an electrical property of an object whichcomprises: an array of sensor elements for placement in electricalcontact with the surface of the object; means for applying a voltage tothe array of sensor elements; means for measuring the electrical chargeat each of the sensor elements while the voltage is applied; means fortransforming the measured electrical charges to a set of transformedsurface charges; a charge correlation matrix which relates transformedsurface charges at the sensor element to internal charges at an internalarray of locations in the object; means for calculating the distributionof charges at the internal array of locations in the object bymultiplying the set of transformed surface charges by the stored chargecorrelation matrix; and means for producing an image from the chargedistribution by calculating the value of an electrical property at eachlocation in the internal array of locations.
 13. The imaging system asrecited in claim 12 in which a plurality of charge correlation matricesare provided, each charge correlation matrix being associated with aselected arrangement of sensor elements and a selected internal array oflocations.
 14. The imaging systems as recited in claim 13 which includesmeans for selecting one of the plurality of charge correlation matricesbased on a prescribed image resolution.